EXPECTED VALUE: ORCHESTRATING UNDERSTANDING Presentation at Palm Springs 11/6/15 Jim

Statistical Inference is Irrefutable!

Take a minute to think about, and then be ready to share with the others at your table:  Name  School  District  Something you really like about the Probability and Statistics in the California CCS-Math  One thing you hope to learn today Introductions 3

 Deepen understanding of expected value – looking at what it means, not the formula for computing it  Engage in hands-on classroom activities designed to develop conceptual understanding of expected value  Special thanks to Sherry Fraser and the other authors of the Interactive Mathematics Program Workshop Goals 4

ATP Administrator Training - Module 1 – MS/HS Math Workshop Norms 1. Bring and assume best intentions. 2. Step up, step back. 3. Be respectful, and solutions oriented. 4. Turn off (or mute) electronic devices.

Guidelines for Assessment and Instruction in Statistics Education (GAISE) Report Statistical problem solving is an investigative process that involves four components: I Formulate Questions – clarify the problem at hand – formulate one (or more) questions that can be answered with data II Collect Data – design a plan to collect appropriate data – employ the plan to collect the data III Analyze Data – select appropriate graphical and numerical methods – use these methods to analyze the data IV Interpret Results – interpret the analysis – relate the interpretation to the original question

Mathematical Modeling What is mathematical modeling? “Modeling is the process of choosing and using appropriate mathematics and statistics to analyze empirical situations, to understand them better, and to improve decisions.” Process: ▫ Identify variables and select those that are essential ▫ Formulate a model to describe the relationships ▫ Analyze and perform operations to draw conclusions ▫ Interpret results in the light of the context ▫ Validate the conclusions ▫ Report on the conclusions and reasoning behind them Importance of Probability and Statistics in K-12 Mathematics

Connecting Math Across Grade Levels # OF PEOPLE LENGTH OF CUBIT (CM) | | | | | | | | | | Grades 3-5 Grades 6-8 High School Mean: 39.3 cm Standard Deviation: 2.2 cm Importance of Probability and Statistics in K-12 Mathematics

Access and Equity The study of statistics offers opportunities for Culturally Responsive Instruction by allowing students to collect and analyze real-world data relevant to their lives The study of statistics requires teachers to attend to issues of language through – Reading – Writing – Listening – Speaking Importance of Probability and Statistics in K-12 Mathematics

Agreeing with Arthur Benjamin  Brief TED talk by Arthur Benjamin:  Arthur Benjamin- Teach statistics before calculus! - Talk Video - TED.com[via torchbrowser.com].flv Arthur Benjamin- Teach statistics before calculus! - Talk Video - TED.com[via torchbrowser.com].flv

Notice and Wonder

Statistical Reasoning Process  Questions  Collect Data  Analyze  Interpret  Is this a standard deck of cards?  Pick one card at a time with replacement and record the results.  Calculate the probabilities  Use the probability to draw your conclusion

Pick a Card! XP(X)Interpretation Black card0.5No big deal

Pick a Card! XP(X)Interpretation Black card0.5No big deal 2 nd Black0.25Still no big deal

Pick a Card! XP(X)Interpretation Black card0.5No big deal 2 nd Black0.25Still no big deal 3 rd Black0.125A little strange, but not unreasonable

Pick a Card! XP(X)Interpretation Black card0.5No big deal 2 nd Black0.25Still no big deal 3 rd Black0.125 A little strange, but not unreasonable 4 th Black Very strange, we wonder, but it’s possible

Pick a Card! XP(X)Interpretation Black card0.5No big deal 2 nd Black0.25Still no big deal 3 rd Black0.125 A little strange, but not unreasonable 4 th Black Very strange, we wonder, but it’s possible 5 th Black We want to check the deck!! The 5% threshold in Statistics is not arbitrary!

Never Tell An Answer Please remember the enormous responsibility we all have as learners not to spoil anybody else’s fun. The quickest way to spoil someone else’s fun is to tell them an answer before they have a chance to discover it themselves. Susan Pirie

Events With Different Values  Do “Rug Games”  What are we using to compute probabilities?  Now do “Pointed Rugs”  How has the previous problem been changed?  Do “Spinner Give and Take”  How are “Pointed Rugs” and “Spinner Give and Take” the same? How are they different?  How could “Spinner Give and Take” be changed to make it “fair”? What makes a game of chance “fair”?

Expected Value  “One-and-One”  Who can explain a “one-and-one” situation in basketball?  What is your intuition about the number of points Terry will make for her team per one-and-one situation in the long run?  Working in groups of 3, at most 4, complete 50 simulations of a “one-and-one” with Terry shooting, and use your data to complete “A Sixty-Percent Solution”  Now create an area model to develop a theoretical analysis of the situation. How many points per situation for Terry in the long run? From the Interactive Mathematics Program: Year 1, The Game of Pig. Copyright © 2009 by IMP, Inc. Used by permission of the publisher, It's About Time,

Conditional Probability  P(A|B) = P B (A) is the probability of A occurring given that B has occurred.  Example:  What is the probability that you will cough at some point today?  What is the probability that you will cough at some point today if you have a cold?  Roll a pair of dice, die G and die H  What is the probability that G = 2?  What is the probability that G = 2 given that G+H≤5?

Conditional Probability

What Have We Done?  Begin with experiences to build a conceptual understanding  Build from there to the formal mathematics  Allow for student agency and authority

Evaluations  Thank you for attending this section  Please take a moment to provide feedback on the session per the next two slides  Suggestions for improvement are welcomed!

Send your text message to this Phone Number: Strongly Disagree Strongly Agree Disagree Agree Speaker was well- prepared and knowledgeable (0-3) Speaker was engaging and an effective presenter (0-3) Session matched title and description in program book (0-3) Other comments, suggestions, or feedback (words) ___ ___ ___ ___________ _10472_ Example: Inspiring, good content poll code for this session (no spaces) Non-Example: Inspiring, good content (1 space) Non-Example: Inspiring, good content